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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubclsubN | Structured version Visualization version GIF version |
Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclsub.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
psubclsub.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
psubclsubN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
2 | psubclsub.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
3 | 1, 2 | psubcli2N 37077 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) = 𝑋) |
4 | eqid 2823 | . . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | 4, 1, 2 | psubcliN 37076 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) = 𝑋)) |
6 | 5 | simpld 497 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾)) |
7 | psubclsub.s | . . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾) | |
8 | 4, 7, 1 | polsubN 37045 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃‘𝐾)‘𝑋) ∈ 𝑆) |
9 | 6, 8 | syldan 593 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥𝑃‘𝐾)‘𝑋) ∈ 𝑆) |
10 | 4, 7 | psubssat 36892 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ((⊥𝑃‘𝐾)‘𝑋) ∈ 𝑆) → ((⊥𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) |
11 | 9, 10 | syldan 593 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) |
12 | 4, 7, 1 | polsubN 37045 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ((⊥𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) ∈ 𝑆) |
13 | 11, 12 | syldan 593 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) ∈ 𝑆) |
14 | 3, 13 | eqeltrrd 2916 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ‘cfv 6357 Atomscatm 36401 HLchlt 36488 PSubSpcpsubsp 36634 ⊥𝑃cpolN 37040 PSubClcpscN 37072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-undef 7941 df-proset 17540 df-poset 17558 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-psubsp 36641 df-pmap 36642 df-polarityN 37041 df-psubclN 37073 |
This theorem is referenced by: pclfinclN 37088 |
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